Geodesic
On the uniqueness of a Walsh series converging on subsequences of partial sum
Matematičeskie zametki, Tome 16 (1974) no. 1, pp. 27-32 Cet article a éte moissonné depuis la source Math-Net.Ru

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We show that if a Walsh series whose coefficients tend towards zero is such that the subsequence of its partial sums indexed by $n_k$, where $n_k$ satisfies the condition $2^{k-1}, tends everywhere, except possibly for a denumerable set, towards a bounded function $f(x)$, then this series is the Fourier series of the function $f(x)$. 
@article{MZM_1974_16_1_a2,
     author = {V. A. Skvortsov},
     title = {On the uniqueness of a {Walsh} series converging on subsequences of partial sum},
     journal = {Matemati\v{c}eskie zametki},
     pages = {27--32},
     year = {1974},
     volume = {16},
     number = {1},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_1974_16_1_a2/}
}
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V. A. Skvortsov. On the uniqueness of a Walsh series converging on subsequences of partial sum. Matematičeskie zametki, Tome 16 (1974) no. 1, pp. 27-32. http://geodesic.mathdoc.fr/item/MZM_1974_16_1_a2/